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Being a valid argument does not necessarily mean the conclusion will be true. It is valid because if the premises are true, then the conclusion has to be true. This can be proven for any valid argument form using a truth table which shows that there is no situation in which there are all true premises and a false conclusion. [2]
Kant disagreed because, he claimed, only categorical judgments are so defined. Hypothetical and disjunctive judgments are a relation between two judgments. In his footnote, Kant asserted that the lengthy and detailed doctrine of the four syllogistic figures concerned only categorical syllogisms or inferences.
Sometimes a syllogism that is apparently fallacious because it is stated with more than three terms can be translated into an equivalent, valid three term syllogism. [2] For example: Major premise: No humans are immortal. Minor premise: All Greeks are people. Conclusion: All Greeks are mortal.
It is clear that nothing would prevent a singular term occurring in a syllogism—so long as it was always in the subject position—however, such a syllogism, even if valid, is not a categorical syllogism. An example is Socrates is a man, all men are mortal, therefore Socrates is mortal.
The study of arguments using categorical statements (i.e., syllogisms) forms an important branch of deductive reasoning that began with the Ancient Greeks. The Ancient Greeks such as Aristotle identified four primary distinct types of categorical proposition and gave them standard forms (now often called A, E, I, and O).
The corresponding conditional of a valid argument is a logical truth and the negation of its corresponding conditional is a contradiction. The conclusion is a necessary consequence of its premises. An argument that is not valid is said to be "invalid". An example of a valid (and sound) argument is given by the following well-known syllogism:
The inability of affirmative premises to reach a negative conclusion is usually cited as one of the basic rules of constructing a valid categorical syllogism. Statements in syllogisms can be identified as the following forms: a: All A is B. (affirmative) e: No A is B. (negative) i: Some A is B. (affirmative) o: Some A is not B. (negative)
categorical syllogism A form of deductive reasoning in Aristotelian logic consisting of three categorical propositions that involve three terms and deduce a conclusion from two premises. category In mathematics and logic, a collection of objects and morphisms between them that satisfies certain axioms, fundamental to category theory. category ...